Solve for `y : (3y-2)/(4) - (2y + 3)/(3) = (2)/(5) - y`.

Options

A. `(23)/(13)`
B. `1(49)/(65)`
C. `(14)/(65)`
D. 3

Answer

To solve the equation \(\frac{3y - 2}{4} - \frac{2y + 3}{3} = \frac{2}{5} - y\), we will follow these steps:

### Step 1: Find a common denominator
The denominators on the left side are 4 and 3. The least common multiple (LCM) of 4 and 3 is 12. We will rewrite each fraction with a denominator of 12.

\[ \frac{3y - 2}{4} = \frac{3(3y - 2)}{12} = \frac{9y - 6}{12} \]
\[ \frac{2y + 3}{3} = \frac{4(2y + 3)}{12} = \frac{8y + 12}{12} \]

Now, we can rewrite the left side of the equation:

\[ \frac{9y - 6}{12} - \frac{8y + 12}{12} = \frac{(9y - 6) - (8y + 12)}{12} = \frac{9y - 6 - 8y - 12}{12} = \frac{y - 18}{12} \]

### Step 2: Rewrite the equation
Now, we can rewrite the original equation using the common denominator:

\[ \frac{y - 18}{12} = \frac{2}{5} - y \]

### Step 3: Eliminate the fraction
To eliminate the fraction, we can multiply both sides of the equation by 60 (the LCM of 12 and 5):

\[ 60 \cdot \frac{y - 18}{12} = 60 \left(\frac{2}{5} - y\right) \]

This simplifies to:

\[ 5(y - 18) = 12(2 - 5y) \]

### Step 4: Distribute and simplify
Now, distribute both sides:

\[ 5y - 90 = 24 - 60y \]

### Step 5: Combine like terms
Next, we will move all terms involving \(y\) to one side and constant terms to the other side:

\[ 5y + 60y = 24 + 90 \]

This simplifies to:

\[ 65y = 114 \]

### Step 6: Solve for \(y\)
Now, divide both sides by 65:

\[ y = \frac{114}{65} \]

### Step 7: Simplify the fraction (if necessary)
The fraction \(\frac{114}{65}\) cannot be simplified further, but we can express it as a mixed number:

\[ y = 1 \frac{49}{65} \]

### Final Answer
Thus, the solution for \(y\) is:

\[ y = \frac{114}{65} \quad \text{or} \quad y = 1 \frac{49}{65} \]

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